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1400 - 1499

1400

14002 = 63 + 653 + 1193.

14002 = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 19)(20 + 21 + 22 + ... + 44),
14002 = (1)(2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10 + ... + 167),
14002 = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 45),
14002 = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 11)(12 + 13)(14 + 15 + 16 + ... + 21),
14002 = 175 x 176 + 176 x 177 + 177 x 178 + 178 x 179 + ... + 223 x 224.

Page of Squares : First Upload January 29, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1401

The 4-by-4 magic squares consisting of different squares with constant 1401:

02142232262
162252222 62
192242 82202
282 22182172
     
22 62202312
72262242102
182252162142
322 82132122

Page of Squares : First Upload February 16, 2010 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1404

481k + 624k + 1404k + 1716k are squares for k = 1,2,3 (652, 23532, 904152).

14042 = (1)(2)(3)(4 + 5)(6)(7 + 8 + 9 + ... + 110),
14042 = (1)(2)(3)(4 + 5 + 6 + ... + 9)(10 + 11 + 12 + ... + 17)(18 + 19 + 20 + 21),
14042 = (1)(2 + 3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + ... + 17)(18 + 19 + 20 + 21),
14042 = (1 + 2)(3)(4)(5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 17)(18),
14042 = (1 + 2)(3)(4 + 5 + 6 + ... + 12)(13)(14 + 15 + 20 + ... + 25),
14042 = (1 + 2)(3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + ... + 28),
14042 = (1 + 2 + 3)(4 + 5)(6)(7 + 8 + 9 + ... + 110),
14042 = (1 + 2 + 3)(4 + 5 + 6 + ... + 9)(10 + 11 + 12 + ... + 17)(18 + 19 + 20 + 21).

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1405

14052 = 1974025, a square with different digits.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1406

The 4-by-4 magic squares consisting of different squares with constant 1406:

02 92132342
112222242152
182212252 42
312202 62 32
     
02 92222292
172 82272182
212302 72 42
262192122152
     
12 32102362
122342 92 52
192152282 62
302 42212 72
     
12102242272
122212252142
192282 62152
302 92132162

Page of Squares : First Upload February 16, 2010 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1407

14072 = 1512 + 1532 + 1552 + 1572 + 1582 + 1592 + ... + 2472.

14072 = 44 + 44 + 184 + 374.

The 4-by-4 magic squares consisting of different squares with constant 1407:

12 22212312
52272222132
152252192142
342 72112 92
     
12 22212312
62352112 52
232132222152
292 32192142
     
12 22212312
92172262192
102332132 72
352 52112 62
     
12 22212312
112352 52 62
142 32292192
332132102 72
12 62232292
112192222212
142312152 52
332 72132102
     
12 92102352
112262212132
142192292 32
332172 52 22

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1408

14082 = (12 + 7)(22 + 7)(112 + 7)(132 + 7) = (12 + 7)(32 + 7)(92 + 7)(132 + 7)
= (12 + 7)(92 + 7)(532 + 7) = (22 + 7)(32 + 7)(92 + 7)(112 + 7) = (22 + 7)(52 + 7)(752 + 7)
= (32 + 7)(112 + 7)(312 + 7) = (92 + 7)(112 + 7)(132 + 7).

Page of Squares : First Upload December 14, 2013 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1410

66k + 210k + 1230k + 1410k are squares for k = 1,2,3 (542, 18842, 683642).

The 4-by-4 magic squares consisting of different squares with constant 1410:

02 12252282
82272192162
112262182172
352 22102 92
   
02 42 52372
132232262 62
202282152 12
292 92222 22
   
02 42132352
162322 92 72
232192222 62
252 32262102
   
02 52192322
72342142 32
202152232162
312 22182112
   
02 52192322
132342 62 72
202152232162
292 22222 92
02 82112352
162122312 72
232292 22 62
252192182102
     
12 52222302
72352 62102
242 42232172
282122192112
     
22 92102352
112282192122
142172302 52
332162 72 42
     
22 92222292
112282212 82
182232142192
312 42172122

Page of Squares : First Upload February 16, 2010 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1411

207417k + 378148k + 632128k + 773228k are squares for k = 1,2,3 (14112, 10878812, 8819780032).

Page of Squares : First Upload April 22, 2011 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1412

14122 = 733 + 803 + 1033.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1413

14132 = 363 + 733 + 1163.

The 4-by-4 magic square consisting of different squares with constant 1413:

02 22252282
102362 42 12
172 82242222
322 72142122

Page of Squares : First Upload July 7, 2008 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1414

14142 = 1999396, a square with odd digits except the last digit 6.

1414 = (12 + 22 + 32 + ... + 2522) / (12 + 22 + 32 + ... + 222).

14142 = 1999396 with 1936 = 442 (1999396 is a square pegged by 9).

Page of Squares : First Upload January 29, 2007 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

1415

14152 = 2002225, a square with jusut 3 kinds of digits : 0, 2 and 5.

The square root of 1415 is 37.61..., 37 = 62 + 12.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1416

A cubic polynomial :
(X + 14162)(X + 15682)(X + 25832) = X3 + 33372X2 + 58915922X + 57350039042.

The 4-by-4 magic square consisting of different squares with constant 1416:

02 82142342
162322 62102
222 22282122
262182202 42

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1417

1 / 1417 = 0.000705716302046577275935...,
72 + 05712 + 6302 + 22 + 0462 + 5772 + 2752 + 9352 = 14172.

14172 = (32 + 4)(3932 + 4).

793k + 1417k + 2977k + 11713k are squares for k = 1,2,3 (1302, 121942, 12793302).

Page of Squares : First Upload January 29, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1418

14182 = 2010724, a zigzag square.

14182± 3 are primes.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1419

14192 = (52 + 8)(2472 + 8).

14195 = 5753233191123099 :
52 + 72 + 52 + 32 + 22 + 32 + 32 + 12 + 92 + 12 + 122 + 302 + 92 + 92 = 1419.

The 4-by-4 magic squares consisting of different squares with constant 1419:

02 52132352
72312202 32
232122252112
292172152 82
     
02 72232292
132352 32 42
172 82252212
312 92162112
     
12 42212312
82352 72 92
252 32232162
272132202112

Page of Squares : First Upload December 8, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1420

14202± 3 are primes.

Page of Squares : First Upload January 16, 2014 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1421

14212 = (137 + 138 + 139 + 140 + 141 + 142 + 143)2 + (144 + 145 + 146 + 147 + 148 + 149 + 150)2.

665k + 1421k + 2401k + 5117k are squares for k = 1,2,3 (982, 58662, 3885702).

The 4-by-4 magic square consisting of different squares with constant 1421:

02 22112362
42212302 82
262242122 52
272202162 62

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1422

14222 = (59 + 60 + 61)2 + (62 + 63 + 64)2 + (65 + 66 + 67)2 + ... + (128 + 129 + 130)2.

14222 = 503 + 933 + 1033.

14222 = 2022084, a square with even digits.

126k + 538k + 1050k + 1422k are squares for k = 1,2,3 (562, 18522, 647362).

The 4-by-4 magic squares consisting of different squares with constant 1422:

02 22 72372
92272242 62
212172262 42
302202112 12
   
02 32182332
92222292 42
212232162142
302202 12112
   
02 32182332
92342132 42
212162232142
302 12202112
   
02 32182332
102252242112
192282 92142
312 22212 42
   
02 32182332
112342 92 82
252162212102
262 12242132
02 92212302
102272232 82
192242142172
312 62162132
   
12 22112362
162312142 32
182212242 92
292 42232 62
   
12 22242292
42352 92102
262 72212162
272122182152
   
12 22242292
52282182172
102252212162
362 32 92 62
   
12 22242292
112322142 92
122132252222
342152 52 42
12 32162342
42222292 92
262202152112
272232102 82
   
12 62192322
162312142 32
182 52282172
292202 92102
   
22 42212312
82342 92112
252 52242142
272152182122
   
32102172322
202152262112
222292 42 92
232162212142
   
42 92222292
112242252102
182272122152
312 62132162

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1423

14232 = 2024929, 2 + 0 * 2 + 49 * 29 = 2 * 0 + 2 + 49 * 29 = 1423.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1425

14252 = 2030625, a zigzag square.

14252 = 253 + 953 + 1053.

The 4-by-4 magic squares consisting of different squares with constant 1425:

02 42252282
102272202142
132262162182
342 22122112
     
02 42252282
102272202142
222262122112
292 22162182
     
12 82242282
122222262112
162292 22182
322 62132142
     
12 82242282
122342 22112
162 32262222
322142132 62

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1426

1 / 1426 = 0.00070126227208..., 72 + 02 + 122 + 62 + 22 + 272 + 202 + 82 = 1426,
1 / 1426 = 0.00070126227208..., 72 + 0122 + 62 + 22 + 272 + 202 + 82 = 1426.

The 5-by-5 magic squares consisting of different squares with constant 1426:

02 32202212242
102162182112252
172302132 82 22
192 62 72282142
262152222 42 52
     
1 2 22112122342
102 52262242 72
142172222212 42
202282 82 32132
272182 92162 62
     
12 52102122342
112262 72242 22
142172222212 42
182 62272162 92
282202 82 32132

Page of Squares : First Upload January 29, 2007 ; Last Revised October 19, 2009
by Yoshio Mimura, Kobe, Japan

1427

The 5-by-5 magic square consisting of different squares with constant 1427:

02 12 32242292
52312132162 42
122192212 92202
232 22222172112
272102182152 72

Page of Squares : First Upload October 19, 2009 ; Last Revised October 19, 2009
by Yoshio Mimura, Kobe, Japan

1428

14282 = 2039184, a square with different digits.

14282 = 263 + 633 + 1213.

The integral triangle of sides 2169, 2897, 4760 (or 313, 43911, 44210) has square area 14282.

14282 = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 12)(13 + 14 + 15)(16 + 17 + 18),
14282 = (1)(2)(3 + 4 + 5 + ... + 53)(54 + 55 + 56 + ... + 65),
14282 = (1 + 2 + 3 + ... + 16)(17)(18 + 19 + 20 + ... + 45),
14282 = (1 + 2 + 3 + ... + 6)(7)(8)(9 + 10 + 11 + ... + 59),
14282 = (1 + 2 + 3 + ... + 6)(7)(8 + 9)(10 + 11 + 12 + ... + 41),
14282 = (1 + 2 + 3 + ... + 6)(7)(8 + 9 + 10 + ... + 24)(25 + 26).

Page of Squares : First Upload January 29, 2007 ; Last Revised October 4, 2011
by Yoshio Mimura, Kobe, Japan

1429

A cubic polynomial :
(X + 842)(X + 8192)(X + 11682) = X3 + 14292X2 + 9640682X + 803537282.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1430

14302 = (1)(2 + 3)(4)(5 + 6)(7 + 8 + 9 + ... + 136),
14302 = (1 + 2 + 3 + 4)(5 + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + 23 + 24).

14302 = (43 + 44)2 + (45 + 46)2 + (47 + 48)2+ (49 + 50)2 + (51 + 52)2 + ...+ (145 + 146)2.

376090k + 439010k + 441870k + 787930k are squares for k = 1,2,3 (14302, 10725002, 8445437002).

The 4-by-4 magic squares consisting of different squares with constant 1430:

02 52262272
72362 92 22
152102232242
342 32122112
     
22 72 92362
122152312102
212302 82 52
292162182 32
     
32 42262272
62172232242
1923021225 2
322152 92102
     
42152172302
182312 92 82
192122222212
272102242 52

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1431

14312 = 13 + 23 + 33 + 43 + 53 + ... + 533.

14312 = 93 + 363 + 1263.

14314 = 4193325113121, and 42 + 12 + 92 + 32 + 322 + 52 + 112 + 32 + 122 + 12 = 1431.

The 4-by-4 magic square consisting of different squares with constant 1431:

12 52 62372
132222272 72
192292152 22
302 92212 32

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1432

14322 is a zigzag square.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1433

14332 = 2053489, a square with different digits.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1434

14342 = S2(57) + S2(181), where S2(n) = 12 + 22 + 32 + ... + n2.

14342 = 93 + 943 + 1073.

The sum of the squares of divisors of 1434 is a square, 16902.

The 4-by-4 magic squares consisting of different squares with constant 1434:

02 12 82372
172222252 62
192302132 22
282 72242 52
     
02 82232292
112272222102
172252142182
322 42152132
     
12 22232302
42272202172
112262212142
362 52 82 72
     
12 22232302
42352122 72
242132202172
292 62192142
12 32202322
72352122 42
222142232152
302 22192132
     
12 82122352
92282202132
142192292 62
342152 72 22

Page of Squares : First Upload January 29, 2007 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan

1435

Komachi equation: 14352 = 1232 * 42 * 52 * 62 * 72 / 82 / 92.

Page of Squares : First Upload July 27, 2010 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1437

14372 is a zigzag square.

94842k + 468462k + 593481k + 908184k are squares for k = 1,2,3 (14372, 11855252, 10304195312).
140826k + 425352k + 547497k + 951294k are squares for k = 1,2,3 (14372, 11855252, 10510692212).

The 4-by-4 magic square consisting of different squares with constant 1437:

02132222282
142262232 62
202242102192
292 42182162

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1439

The 5-by-5 magic square consisting of different squares with constant 1439:

02 12 52182332
62292162 92152
72232262132 42
252 82192172102
272 22112242 32

Page of Squares : First Upload October 19, 2009 ; Last Revised October 19, 2009
by Yoshio Mimura, Kobe, Japan

1440

14402 = (22 - 1)(172 - 1)(492 - 1) = (22 - 1)(32 -1)(42 - 1)(72 - 1)(112 - 1)
= (22 - 1)(42 - 1)(72 - 1)(312 - 1) = (22 - 1)(52 - 1)(92 - 1)(192 - 1) = (32 - 1)(42 - 1)(72 - 1)(192 - 1)
= (42 - 1)(52 - 1)(72 - 1)(112 - 1) = (72 - 1)(112 - 1)(192 - 1) = (92 - 1)(1612 - 1).

14402 = 103 + 893 + 1113 = 403 + 963 + 1043.

14402 = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9)(10 + 11 + 12 + 13 + 14)(15 + 16 + 17),
14402 = (1 + 2 + 3 + ... + 15)(16)(17 + 18 + 19)(20).

Cubic polynomials :
(X + 6002)(X + 14402)(X + 15472) = X3 + 21972X2 + 25633202X + 13366080002,
(X + 14402)(X + 16122)(X + 58592) = X3 + 62452X2 + 128752922X + 136003795202.

The quadratic polynomial -1440X2 + 10920X - 9191 takes the values 172, 832, 1032, 1072, 972, 672 at X = 1, 2,..., 6,

Page of Squares : First Upload January 29, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1441

14412 = 2076481, a square with different digits.

14412 = 93 + 963 + 1063.

Page of Squares : First Upload January 29, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1442

14422 = 2079364, a square with different digits.

14422 = 2079364 with 207936 = 4562 and 4 = 22.

64890k + 232162k + 803194k + 979118k are squares for k = 1,2,3 (14422, 12891482, 12122692122).

The 4-by-4 magic squares consisting of different squares with constant 1442:

02 52242292
152 22272222
162332 42 92
312182112 62
     
12 92242282
102182272172
212292 42122
302142112152

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1443

14432 = 72 + 262 + 452 + 642 + 832 + 1022 + 1212 + ... + 4822.

The 4-by-4 magic square consisting of different squares with constant 1443:

02 52 72372
112272232 82
192172282 32
312202 92 12

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1444

The square of 38.

14442 = 2085136, a square with different digits.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1445

14452 = (43 + 44 + 45 + ... + 59)2 + (60 + 61 + 62 + ... + 76)2.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1446

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1446 - 2312 - 673 - 5365 - ... - 5114 - 2797 - 10138 - 1446
(Note f(1446) = 142 + 462 = 2312,   f(2312) = 232 + 122 = 673, etc. See 37)

14462 = 2090916, a zigzag square.

The 4-by-4 magic squares consisting of different squares with constant 1446:

12 42232302
82272222132
152262172162
342 52122112
     
12 42232302
82352 62112
152142252202
342 32162 52
     
12 72102362
142202272112
152312162 22
322 62192 52
     
42132222282
142242252 72
202262 92172
292 52162182

The 5-by-5 magic square consisting of different squares with constant 1446:

02 12122252262
52362 32 42102
192 82162182212
222 62292 92 22
242 72142202152

Page of Squares : First Upload January 29, 2007 ; Last Revised February 16, 2010
by Yoshio Mimura, Kobe, Japan

1447

14472 = 20903809 is a zigzag square.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1448

14482 = 2096704 is a zigzag square.

14482 = 523 + 783 + 1143.

Page of Squares : First Upload January 29, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1449

A cubic polynomial :
(X + 10562)(X + 11482)(X + 14492) = X3 + 21292X2 + 25647722X + 17566053122.

14492 = 403 + 843 + 1133.

14492 = (1 + 2)(3 + 4 + 5 + ... + 11)(12 + 13 + 14 + ... + 149).

14492 = (42 + 5)(82 + 5)(382 + 5).

14495 = 6387661996482249 : 62 + 32 + 82 + 72 + 62 + 62 + 192 + 92 + 62 + 42 + 82 + 22 + 242 + 92 = 1449.

The 4-by-4 magic squares consisting of different squares with constant 1449:

02 32122362
152322142 22
182202252102
302 42222 72
     
02 32122362
172162302 22
222282 92102
262202182 72
     
02 62182332
122342 72102
242 12262142
272162202 82
     
02122242272
152262222 82
182252102202
302 22172162

Page of Squares : First Upload January 29, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1450

14502 = 683 + 793 + 1093.

14502 = (22 + 1)(32 + 1)(122 + 1)(172 + 1) = (52 + 4)(112 + 4)(242 + 4).

1450k + 12818k + 24186k + 45646k are squares for k = 1,2,3 (2902, 532442, 105528682).

The 5-by-5 magic squares consisting of different squares with constant 1450:

02 42112172322
72132202242162
102302 92152122
252 22282 62 12
262192 82182 52

Page of Squares : First Upload July 7, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1451

14512 = 2105401, 210 * 5 + 401 = 1451.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1452

14522 = (52 + 8)(62 + 8)(382 + 8).

14522 = 773 + 883 + 993.

Page of Squares : First Upload July 7, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1455

14552 = 23 + 733 + 1203 = 253 + 873 + 1133.

Komachi equation: 14552 = 92 / 82 / 72 / 62 * 54322 * 102.

The 4-by-4 magic squares consisting of different squares with constant 1455:

12 22192332
32292222112
172212232142
342132 92 72
     
12 22192332
32292222112
172232212142
342 92132 72
     
12 32222312
72352 92102
262 52232152
272142192132

The 5-by-5 magic squares consisting of different squares with constant 1455:

02 12102252272
72282172 32182
112212292 62 42
142152122232192
332 22 92162 52

Page of Squares : First Upload July 7, 2008 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1456

14562 = (12 + 3)(52 + 3)(72 + 3)(192 + 3).

Page of Squares : First Upload December 14, 2013 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1457

14572 = 183 + 733 + 1203.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1458

14582 = 134 + 134 + 214 + 374 = 274 + 274 + 274 + 274 = 95 + 95 + 95 + 95 + 185 = 96 + 96 + 96 + 96.

14582 = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 49),
14582 = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 13)(14 + 15 + 16 + ... + 40),
14582 = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 31).

The 4-by-4 magic squares consisting of different squares with constant 1458:

02 82132352
122322172 12
152192262142
332 32182 62
     
02122152332
162322 32132
192 12302142
292172182 22
     
12 62142352
162332 72 82
242 32272122
252182222 52
     
12122232282
142272222 72
192242112202
302 32182152
22 62 72372
132212282 82
182302152 32
312 92202 42
     
32 52202322
62302212 92
182222192172
332 72162 82

Page of Squares : First Upload January 29, 2007 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1459

14592 = 30 + 36 + 37 + 312 + 313.

Page of Squares : First Upload August 29, 2011 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

1460

170k + 370k + 830k + 1130k are squares for k = 1,2,3 (502, 14602, 455002).

Page of Squares : First Upload April 22, 2011 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1461

311193k + 400314k + 487974k + 935040k are squares for k = 1,2,3 (14612, 11702612, 10138974752).

The 4-by-4 magic square consisting of different squares with constant 1461:

22122232282
142252242 82
192262102182
302 42162172

Page of Squares : First Upload March 1, 2010 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1462

14622 = 2137444, 2 + 13 * 7 * 4 * 4 + 4 = 1462.

S2(1462) = S2(5) x S2(6) x S2(85), where S2(n) = 12 + 22 + 32 + ... + n2.

14622 = 393 + 503 + 1253.

Page of Squares : First Upload January 29, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1463

14632 = 2140369, a square with different digits.

14632 = (21 + 22 + 23 + 24 + 25 + 26 + 27)2 + (28 + 29 + 30 + 31 + 32 + 33 + 34)2 + (35 + 36 + 37 + 38 + 39 + 40 + 41)2 + ... + (91 + 92 + 93 + 94 + 95 + 96 + 97)2.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1464

14642 = 243 + 713 + 1213 = 723 + 953 + 973.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1466

1 / 1466 = 0.0006821282401..., 62 + 82 + 22 + 12 + 282 + 242 + 02 + 12 = 1466,
1 / 1466 = 0.0006821282401..., 62 + 82 + 22 + 12 + 282 + 242 + 012 = 1466.

14662 = 493 + 993 + 1023 = 18 + 28 + 38 + 38 + 48 + 58 + 68.

Page of Squares : First Upload January 29, 2007 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1467

14672 = (1 + 2 + 3 + ... + 162)(163).

The 4-by-4 magic square consisting of different squares with constant 1467:

12 42192332
152312162 52
202 72272172
292212112 82

Page of Squares : First Upload January 29, 2007 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1468

14682 = 44 + 84 + 164 + 384.

Page of Squares : First Upload July 7, 2008 ; Last Revised July 7, 2008
by Yoshio Mimura, Kobe, Japan

1469

399568k + 430417k + 593476k + 734500k are squares for k = 1,2,3 (14692, 11120332, 8653423612).

Page of Squares : First Upload April 22, 2011 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1470

14702 = 353 + 493 + 1263.

14702 = (12 + 6)(22 + 6)(62 + 6)(272 + 6) = (12 + 6)(32 + 6)(62 + 6)(222 + 6)
= (122 + 6)(1202 + 6) = (22 + 6)(32 + 6)(1202 + 6) = (62 + 6)(82 + 6)(272 + 6).

210k + 1470k + 14490k + 27930k are squares for k = 1,2,3 (2102, 315002, 49833002).
994k + 1470k + 2338k + 4802k are squares for k = 1,2,3 (982, 56282, 3573082).
430k + 740k + 1470k + 1585k are squares for k = 1,2,3 (652, 23252, 874252).

14702 = (1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + ... + 31),
14702 = (1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + ... + 23),
14702 = (1)(2 + 3)(4 + 5 + 6 + ... + 24)(25 + 26 + 27 + ... + 59),
14702 = (1)(2 + 3)(4 + 5 + 6 + ... + 31)(32 + 33 + 34 + ... + 52),
14702 = (1)(2 + 3 + 4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15 + 22),
14702 = (1)(2 + 3 + 4 + ... + 8)(9)(10)(11 + 12 + 13 + ... + 38),
14702 = (1)(2 + 3 + 4 + ... + 8)(9 + 10 + 11 + ... + 351),
14702 = (1)(2 + 3 + 4)(5 + 6 + 7 + 8 + 9)(10)(11 + 12 + 13 + ... + 38),
14702 = (1 + 2)(3 + 4 + 5 + ... + 9)(10)(11 + 12 + 13 + ... + 59),
14702 = (1 + 2 + 3 + ... + 14)(15 + 16 + 17 + ... + 20)(21 + 22 + 23 + ... + 28),
14702 = (1 + 2 + 3 + ... + 24)(25 + 26 + 27 + ... + 122),
14702 = (1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 13)(14)(15 + 16 + ... + 20).

14702 = 309 x 310 + 311 x 312 + 313 x 314 + 315 x 316 + ... + 347 x 348.

The 4-by-4 magic squares consisting of different squares with constant 1470:

02 12102372
22192332 42
252282 52 62
292182162 72
   
02 12102372
52282252 62
222192242 72
312182132 42
   
02 22252292
52332102162
222192242 72
312 42132182
   
02 22252292
52362102 72
222 12272162
312132 42182
   
02 52222312
132302 12202
252 42272102
262232162 32
12 22132362
32202312102
262252122 52
282212142 72
   
12 22212322
82352102 92
262152202132
272 42232142
   
12 22212322
162332102 52
222192202152
272 42232142
   
12 62 82372
122192312 22
222282112 92
292172182 42
   
12 82262272
102212202232
122312132142
352 22152 42
12 82262272
112222242172
182292 72162
322 92132142
   
12 82262272
122312132142
222112242172
292182 72162
   
12 82262272
182312132 42
192 22242232
282212 72142
   
12112182322
132242232142
202222192152
302172162 52
   
12112182322
132332 42142
202 22292152
302162172 52
12122222292
132262242 72
202252112182
302 52172162
   
22 42192332
72212282142
242222172112
292232 62 82
   
22 42192332
82342152 52
212172222162
312 32202102
   
22 42192332
152 52322142
202302 72112
292232 62 82
   
22 82212312
112 92282222
162342 72 32
332132142 42
32 42222312
72302202112
162232192182
342 52152 82
     
52 62252282
102232202212
162292182 72
332 82112142

The 5-by-5 magic squares consisting of different squares with constant 1470:

02 22 92192322
52202262122152
62 72172302142
252212182 82 42
282242102 12 32
   
02 22 72112362
132242 62252 82
152282142162 32
202 52302122 12
262 92172182102
   
02 52162172302
102 82242272 12
152342 22 62 72
192122252 42182
282 92 32202142
   
02 22 72242292
92302202 52 82
112142272102182
222192162122152
282 32 62252 42

Page of Squares : First Upload January 29, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1471

The 5-by-5 magic squares consisting of different squares with constant 1471:

02 92152182292
142352 52 32 42
172 82312112 62
192 12 22242232
252102162212 72
     
02 12192222252
92142172112282
152312162 52 22
182132232202 72
292122 62212 32
     
02 52142172312
92222232 42192
182272 72152122
212 82112292 22
252132242102 12
     
02 52142172312
92 42292222 72
182232162 12192
212262132112 82
252152 32242 62
02 52142172312
92 42292222 72
182232192 12162
212262 82112132
252152 32242 62
     
02 32 72182332
62 12232292 82
92312162132 22
252102212 42172
272202142112 52

Page of Squares : First Upload October 28, 2009 ; Last Revised October 28, 2009
by Yoshio Mimura, Kobe, Japan

1472

14722 = (12 + 7)(192 + 7)(272 + 7) = (12 + 7)(32 + 7)(42 + 7)(272 + 7)
= (12 + 7)(42 + 7)(52 + 7)(192 + 7) = (32 + 7)(52 + 7)(652 + 7) = (42 + 7)(112 + 7)(272 + 7).

A cubic polynomial :
(X + 5042)(X + 9992)(X + 14722) = X3 + 18492X2 + 17223122X + 7411461122.

14722 = 84 + 164 + 324 + 324.

Page of Squares : First Upload January 29, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1473

14732 = 2169729, 22 + 12 + 62 + 92 + 72 + 22 + 92 = 162.

The 4-by-4 magic square consisting of different squares with constant 1473:

52 82222302
102242262112
182282132142
322 72122162

Page of Squares : First Upload January 29, 2007 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1474

14742± 3 are primes.

The 4-by-4 magic square consisting of different squares with constant 1474:

12 22102372
172262222 52
202132292 82
282252 72 42

The 5-by-5 magic squares consisting of different squares with constant 1474:

02 32 62232302
92 82292 22222
122252202162 72
152262142192 42
322102 12182 52
   
02 12 72202322
62262132232 82
152 32302182 42
222282102 52 92
272 22162142172
   
02 32 62232302
92 82292222 22
122252202 42172
152262142112162
322102 12182 52
   
02 52152182302
82262 22172212
132 12282222 62
202242192112 42
292142102162 92

Page of Squares : First Upload October 28, 2009 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1475

14752 = 2175625, a zigzag square.

14752 = 2175625, 22 + 12 + 72 + 52 + 62 + 22 + 52 = 122.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1476

14762± 5 are primes.

14762 = 204 + 204 + 304 + 324.

14762 = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 36)(37 + 38 + 39 + ... + 45).

14762 = (12 + 8)(22 + 8)(1422 + 8) = (102 + 8)(1422 + 8) = (22 + 8)(192 + 8)(222 + 8).

Page of Squares : First Upload January 29, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1477

14772 = 2181529, a zigzag square.

A cubic polynomial :
(X + 5122)(X + 8372)(X + 11042) = X3 + 14772X2 + 11649122X + 4731125762.

125545k + 186102k + 773948k + 1095934k are squares for k = 1,2,3 (14772, 13603172, 13372772772).
206780k + 454916k + 728161k + 791672k are squares for k = 1,2,3 (14772, 11860312, 9925956952).

14772 = 2181529 appears in the decimal expressions of e:
  e = 2.71828•••2181529••• (from the 15006th digit)
  (2181529 is the sixth 7-digit square in the expression of e.)

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1478

1 / 1478 = 0.000676..., 676 = 262.

226k + 610k + 822k + 1478k are squares for k = 1,2,3 (562, 18122, 634242).

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1479

1 / 1479 = 0.000676..., 676 = 262.

The 4-by-4 magic square consisting of different squares with constant 1479:

12102172332
192292 92142
212232222 52
262 32252132

The 5-by-5 magic squares consisting of different squares with constant 1479:

02 22 32252292
72 12332182 42
152272 82192102
232132142122212
262242112 52 92

Page of Squares : First Upload January 29, 2007 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1480

Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1480 - 6596 - 13441 - 2838 - ... - 6833 - 5713 - 3418 - 1480
(Note f(1480) = 142 + 802 = 6596,   f(6596) = 652 + 962 = 13441, etc. See 1268)

14802 = (12 + 4)(22 + 4)(2342 + 4) = (62 + 4)(2342 + 4).

Page of Squares : First Upload October 9, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1481

14815 = 7124842897431401 : 72 + 122 + 42 + 82 + 42 + 22 + 82 + 92 + 72 + 42 + 312 + 42 + 02 + 12 = 1481.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

1482

1482k + 2850k + 32718k + 79914k are squares for k = 1,2,3 (3422, 864122, 233538122).
1482k + 6630k + 15418k + 19734k are squares for k = 1,2,3 (2082, 259482, 34124482).

The 4-by-4 magic squares consisting of different squares with constant 1482:

12 42212322
62292222112
172242192162
342 72142 92
   
12 62222312
82352 72122
242 52252162
292142182112
   
12 82242292
142352 526 2
182 72252222
312122162112
   
12102152342
162 92322112
212262132142
282252 82 32
   
12102152342
162332 42112
212 22292142
282172202 32

Page of Squares : First Upload March 1, 2010 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1484

14842 = 73 + 173 + 1303.

14845 = 7197298330471424 : 72 + 192 + 72 + 292 + 82 + 32 + 32 + 02 + 42 + 72 + 12 + 42 + 22 + 42 = 1484.

Page of Squares : First Upload July 7, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

1485

14852 = 2205225, a square with just 3 kinds of digits.

14852 = 65 + 95 + 125 + 185.

14852 = (1)(2 + 3)(4 + 5 + 6 + ... + 14)(15 + 16 + 17 + ... + 95),
14852 = (1)(2 + 3 + 4 + ... + 16)(17 + 18 + 19 + ... + 181),
14852 = (1)(2 + 3 + 4)(5 + 6)(7 + 8)(9)(10 + 11 + 12 + ... + 20),
14852 = (1)(2 + 3 + 4)(5 + 6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + ... + 33),
14852 = (1)(2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11)(12 + 13 + 14 + ... + 33),
14852 = (1 + 2)(3 + 4 + 5 + ... + 1212),
14852 = (1 + 2)(3 + 4 + 5 + ... + 24)(25 + 26 + 27 + ... + 74),
14852 = (1 + 2)(3 + 4 + 5 + ... + 8)(9)(10 + 11 + 12 + ... + 15)(16 + 17),
14852 = (1 + 2)(3 + 4 + 5 + ... + 8)(9)(10 + 11 + 12)(13 + 14 + 15 + 16 + 17),
14852 = (1 + 2)(3 + 4 + 5 + ... + 8)(9 + 10 + 11 + 12 + 13)(14 + 15 + 16 + ... + 31).

14852 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + ... + 543.

14852 + 14862 + 14872 + ... + 15122 = 15132 + 15142 + 15152 + ... + 15392.

The 4-by-4 magic squares consisting of different squares with constant 1485:

02 32242302
42122292222
52362 82102
382 62 22 12
   
02 32242302
52362 82102
262 62222172
282122192142
   
02 42 52382
122272242 62
212162282 22
302222102 12
   
02 52262282
122 62272242
212322 42 22
302202 82112
   
02102192322
122342 42132
212 22282162
302152182 62
22 42212322
82262242132
112282182162
362 32122 62
   
22 62172342
112322142122
242 82262132
282192182 42
   
22 62172342
112322182 42
242 82262132
282192142122
   
22 82112362
162132322 62
212242182122
282262 42 32
   
32 42262282
142302172102
162132222242
322202 62 52
42 82262272
132162222242
202292102122
302182152 62
     
42122222292
132242262 82
202272102162
302 62152182
     
42132202302
162 82292182
222262102152
272242122 62

Page of Squares : First Upload January 29, 2007 ; Last Revised September 9, 2011
by Yoshio Mimura, Kobe, Japan

1486

The 5-by-5 magic squares consisting of different squares with constant 1486:

02 72192202262
102252 62142232
162182 12282112
172 22322 52122
292222 82 92 42
     
02 72132222282
102192262 52182
162322142 12 32
172 62212242122
292 42 22202152
     
02 32152242262
72 82322 52182
132362 42 22 12
222 92102252142
282 62112162172
     
02 62152212282
72252182222 22
192142 82172242
202102272162 12
262232122 42112
02 62152212282
122232 42262112
172142192 82242
182252222 72 22
272102202162 12
     
12 22112242282
62212302102 32
82132202232182
192262 72162122
322142 42 52152

Page of Squares : First Upload November 10, 2009 ; Last Revised November 10, 2009
by Yoshio Mimura, Kobe, Japan

1487

14872 = 84 + 144 + 174 + 384.

1 / 1487 = 0.0006724..., 6724 = 822.

14872 = 30 + 31 + 33 + 34 + 38 + 39 + 310 + 312 + 313.

Page of Squares : First Upload January 29, 2007 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

1488

14882 = 2214144, a square with just 3 kinds of digits 1, 2 and 4.

1488k + 2356k + 9145k + 11036k are squares for k = 1,2,3 (1552, 146012, 14578372).

Page of Squares : First Upload January 29, 2007 ; Last Revised April 22, 2011
by Yoshio Mimura, Kobe, Japan

1489

14892 = 2217121, a square with just 3 kinds of digits 1, 2 and 7.

14892 = 2217121, 22 + 22 + 12 + 72 + 12 + 22 + 12 = 82.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1490

S2(417) + S2(1479) = S2(1490), where S2(n) = 12 + 22 + ... + n2.

The 5-by-5 magic square consisting of different squares with constant 1490:

02 52 62232302
92342122102 32
152142182132242
202 72192262 22
282 82252 42 12

Page of Squares : First Upload January 29, 2007 ; Last Revised November 10, 2009
by Yoshio Mimura, Kobe, Japan

1491

14913 = 3314613771, and 32 + 32 + 12 + 42 + 62 + 12 + 372 + 72 + 12 = 1491.

14912 = 2223081 appears in the decimal expressions of π:
  π = 3.14159•••2223081••• (from the 45924th digit)

The 4-by-4 magic squares consisting of different squares with constant 1491:

02 12112372
132282232 32
192252212 82
312 92202 72
     
02 12112372
172322132 32
192212252 82
292 52242 72

Page of Squares : First Upload December 1, 2008 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1492

14922 = 2226064, a square with even digits.

14922 = 2226064, 22 + 22 + 22 + 62 + 02 + 62 + 42 = 102.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1494

The 4-by-4 magic squares consisting of different squares with constant 1494:

02 12 72382
102332172 42
132202302 52
352 22162 32
     
02 12 72382
182232252 42
212302122 32
272 82262 52
     
02 22112372
172 92322102
232252182 42
262282 52 32
     
02 72222312
92282232102
182252162172
332 62152122
02 92182332
132342 52122
222 12282152
292162192 62
     
02102132352
182322 52112
212 32302122
272192202 22
     
12 22202332
82352132 62
232162222152
302 32212122
     
12 22202332
152322142 72
222 52272162
282212132102
22112122352
192162292 62
202212222132
272262 52 82
     
32102192322
122332 62152
212 42292142
302172162 72
     
42142212292
182322112 52
232152162222
252 72262122

The 5-by-5 magic square consisting of different squares with constant 1494:

02 52132202302
112102342 92 62
162272122 22192
212 82 32282142
262242 42152 12

Page of Squares : First Upload November 10, 2009 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1495

1495 = (12 + 22 + 32 + ... + 10002) / (12 + 22 + 32 + ... + 872).

14952 = S2(144) + S2(154), where S2(n) = 12 + 22 + ... + n2.

Komachi equations:
14952 = 12 * 232 / 42 * 52 * 62 * 782 / 92 = 12 * 232 * 452 / 62 * 782 / 92.

14952 = (22 + 23 + 24 + ... + 44)2 + (45 + 46 + 47 + ... + 67)2.

The 5-by-5 magic squares consisting of different squares with constant 1486:

02 32 62192332
92 22232252162
152312 82142 72
172112292122102
302202 52132 12
     
02 12182212272
32202312 22112
62232 42252172
152222132192162
352 92 52 82102
     
02 12182212272
32222 42252192
62202292132 72
152232172142162
352 92 52 82102

Page of Squares : First Upload January 29, 2007 ; Last Revised July 27, 2010
by Yoshio Mimura, Kobe, Japan

1496

14962 = S2(116) + S2(172), where S2(n) = 12 + 22 + ... + n2.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1497

The 4-by-4 magic square consisting of different squares with constant 1497:

02 22 72382
162232262 62
202302142 12
292 82242 42

Page of Squares : First Upload March 1, 2010 ; Last Revised March 1, 2010
by Yoshio Mimura, Kobe, Japan

1498

14982 = 2244004, a square with just 3 kinds of even digits 0, 2 and 4.

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan

1499

(14992 + 7) = (42 + 7)(52 + 7)(62 + 7)(82 + 7).

Page of Squares : First Upload January 29, 2007 ; Last Revised January 29, 2007
by Yoshio Mimura, Kobe, Japan